Date: Nov 17, 2012 11:55 PM
Author: Graham Cooper
Subject: Re: A HARD FLAW in Godel's Proof

On Nov 18, 1:10 pm, "INFINITY POWER" <infin...@limited.com> wrote:
> THINKING CAPS ON!
> ARGUE LOGICALLY!
> ASSUME ANYTHING!
> ROLLBACK ASSUMPTIONS LATER ON <<!
>
> STEP 1:  DEFINE a 2 parameter predicate DERIVE(THEOREM, DERIVATION)
>
> DERIVE(T,D) is TRUE IFF
>   D contains a sequence of inference rules and substitutions
>   and the final formula T in D is logically implied from the Axioms.
>
> - - - - - - - - - - - - - -
>
> STEP 2:  DEFINE a Godel Statement.
>
> i.e.  Godel Statement named G =
>     ALL(M)  ~DERIVE(G,M)
>
> - - - - - - - - - - - - - -
>
> STEP 3:  IS G A THEOREM?
>
> ASSUME: YES G IS A THEOREM
>     DERIVE(   G:ALL(M)~DERIVE(G,M)  , D )
>
> - - - - - - - - - - - - - -
>
> STEP 4:  UNIFY THE QUERY TO THE AXIOMS TO GET THE ANSWER
>
>   GOAL :       DERIVE(  G:ALL(M)~DERIVE(G,M)   , D )
>   SUBGOAL :    G:ALL(M)~DERIVE(G,M)
>
>   (SUBGOALs are a Derivation Process that calculate reverse D in the trace)
>
> - - - - - - - - - - - - - - -
>
> STEP 5:  REMOVE THE QUANTIFIER
>
>   G:~EXIST(M)DERIVE(G,M)
>   G: ~DERIVE(G,M)
>
> M is a variable and Existential by Double Variable Instantiation Rule of
> UNIFY().
>
> - - - - - - - - - - - - - - -
>

INSERT  A STEP:

STEP 6a

G: ~DERIVE(G,  [G | M] )

G is the HEAD of M by definition.  (either 1st or last element)
M are the REMAINING TAIL of deductions back to the axioms.

[G <- <M>]
[G <- N <- <O>]
...
[G <- N <- P <- ... <- AXIOMS ]

Now M is strictly FREE as it doesn't contain G as an element in it's
deduction list.

and 6a reduces to 6 below.




>
> STEP 6:  M IS A FREE VARIABLE
>
>   G: ~DERIVE(G,M)
>
> is a null statement that will return
>
> SUBGOAL:  M?
>
> i.e. When parsed by a clever logic compiler, Godel's Statement will return a
> Query in response
>
> [PROVER]- "Why is sentence G not derivable?"
>
> Herc
>
> --
> if( if(t(S),f(R)) , if(t(R),f(S)) ).
>     if it's sunny then it's not raining
> ergo
>        if it's raining then it's not sunny